# Evaluation of piezoelectric energy harvester under dynamic bending by means of hybrid mathematical/isogeometric analysis

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## Abstract

In this paper, a novel hybrid mathematical/isogeometric analysis (IGA) scheme is implemented to evaluate the energy harvesting of the piezoelectric composite plate under dynamic bending. The NURBS-based IGA is applied to obtain the structural response exerted by the mechanical loading. The dynamic responses conveniently coupled with the governing voltage differential equations to estimate the energy harvested. The capabilities of the scheme are shown with the comparison against analytical and full electromechanical finite element results. As there is no need of fully coupled electromechanical element, the scheme provides cheaper computational cost and could be implemented with standard computational software. Thus, it gives great benefit for early design stage. Moreover, the robustness of the scheme is shown by the couple with high order IGA element which has been proven less prone to the shear locking phenomena in the literature. The computational results show greater accuracy on structural responses and energy estimation for a very thin plate compared to the couple with standard finite element method.

## Keywords

Piezoelectric Energy harvester Dynamic bending Non-uniform rational B-spline (NURBS) Isogeometric analysis (IGA)## 1 Introduction

In this paper, work on piezoelectric energy harvesting via structural vibration is presented. The numbers of the mathematical/computational model on the piezoelectric energy harvesting has been significantly grown in the past decade. One of the earliest mathematical models is a cantilevered piezoelectric under base excitation by Erturk and Inman (2008). The model provided the basis for voltage/energy harvested from a piezoelectric composite structure under mechanical vibration and has been validated with experimental results (Erturk and Inman 2009).

On the piezoelectric energy harvesting, one of the particular fields that attract constant attention is the analysis of flow-induced/aeroelastic vibration as the source of excitation (piezoaeroelastic energy harvesting). The articles by Abdelkefi (2016) and Li et al. (2016) reviewed numerous piezoaeroelastic energy harvesting models developed within the period 2000’s–2015. It can be seen from those reviews, most of the proposed mathematical/computational models were flutter-based harvester for micro-scale application. In addition, apart from the source of excitation, the modeling of the piezoelectric energy harvester nonlinear behavior is also studied in some articles (Pasharavesh et al. 2016; Stanton et al. 2010).

Flutter-based vibration attracts considerable attention due to its self-sustained oscillatory characteristic, thus providing more sustainable power output (Erturk et al. 2010). However, flutter vibration is known for its catastrophic nature. Hence it is considered not applicable as the excitation source for a general use public structure, such as civil structure or aerospace structure, i.e. bridge, transport aircraft.

In contrast with the large development of piezo-aeroelastic energy harvesting model, the analysis under normal operational loads such as low-speed wind load on civil structure or cruise and gust load on aircraft structure has received little attention. To the authors’s knowledge, there are only a few articles proposed models for gust load on wing structure (De Marqui Jr. and Maria 2010; Xiang et al. 2015; Tsushima and Su 2016; Bruni et al. 2017) . Akbar and Curiel-Sosa (2016) proposed the model for a more general condition, dynamic bending load. The model has been successfully implemented on a typical transport aircraft wingbox structure under dynamic cruise load.

Akbar and Curiel-Sosa (2016) proposed a hybrid scheme, in which conveniently coupled the piezoelectric beam voltage equation with the dynamic responses from FEM and analytical model. By means of this scheme, it is possible to implement the responses from the various computational model (not limited to finite element model) as input to the voltage equation to provide the power output estimation.

In this current paper, a novel hybrid mathematical/isogeometric analysis (IGA) scheme is presented. The main attribute of NURBS-based IGA is the ability to establish numerical engineering analysis within the same model from the computational engineering design/drawing. Thus, decreasing the cost of numerical analysis-design interface and providing more efficient computational process compared to standard finite element (Hughes et al. 2005).

The first article of IGA by Hughes et al. (2005) was followed by the development of IGA in structural vibration (Cottrell et al. 2006) and fluid-structure interactions (Bazilevs et al. 2008) which demonstrated an excellent integration with CAD software. The smoothness of NURBS-basis functions has shown a great flexibility on various fluid structure-interaction problems (Bazilevs et al. 2008). On the analysis of structural vibrations, Cottrell et al. (2006) shows that high-order NURBS-based IGA elements provided more accurate frequency spectra than standard high-order finite elements.

A unique *k*-refinement also exists in NURBS-IGA formulation. This strategy involves degree elevation and knot insertion to refine the spaces (Hughes et al. 2005). Standard *p*-refinement as in FEM adds many degree of freedoms to the overall system, while *k*-refinement add fewer degree of freedoms yet still manages to obtain a higher order functions. It is also found that the *k*-version of IGA has better approximation properties per degree of freedom compared to the standard FEM (Evans et al. 2009).

In addition, the ease to construct high order and continuous basis functions provided interesting alternatives on solving high-order PDE problems, i.e., IGA for high-order gradient elasticity (Fischer et al. 2011; Khakalo and Niiranen 2017). Moreover, it is also found that higher-order NURBS-based IGA elements are less prone to the effect of mesh distortion (Lipton et al. 2010). Interested readers are referred to the article by Nguyen et al. (2015) for more detailed review of advantageous and disadvantageous/limitation of IGA.

In this paper, the advantage of IGA against shear locking phenomena on a thin shells and its effect on the energy harvesting response is focused. A study by Thai et al. (2012) shown that high-order IGA elements are hardly suffer from shear locking phenomena. The shear locking phenomena have been investigated since the early development of finite elements. It happens when the shear energy becomes very dominant compared to the bending energy as the thickness of the element is very small compared to its length (Kwon and Bang 2000).

Thai et al. (2012) investigated the benefits of NURBS-based IGA implementation for the structural dynamic problems of the plate with various thickness. At very thin configuration, it was concluded that the high-order IGA elements show more resistance to the shear locking phenomena compares to the standard finite element. This attribute provides an advantage in the modelling of piezoelectric structures which commonly manufactured as thin-walled structures/plates.

As reviewed by Nguyen et al. (2015), the smoothness of NURBS basis function, plate and shell elements are more convenient to be constructed. Therefore, the development of IGA shell elements has attracted significant attention in the recent years. Application on various structural mechanic problems and combination with other methods are found in the literature. Combination of IGA with the level set method is utilized for the analysis of complicated cutout plate (Yin et al. 2015). The use of simple first shear deformation theory (FSDT) with IGA provides a shear-locking free formulation (Yu et al. 2015). Buckling and free vibration analysis of the functionally graded material (FGM) plate (Yin et al. 2017). Buckling exerted by thermal and mechanical loads combination (Yu et al. 2017). A combination of extended IGA (XIGA) and level set method for analysis on a plate with internal defects (Yin et al. 2016). In addition, free-locking high order 3D IGA element also has been developed (Lai et al. 2017a) recently.

Over the past few decades, there have been extensive studies FEM model of piezoelectric structures. One of the earliest work was reported by Allik and Hughes (1970). For the first time, the piezoelectric effect incorporated in a finite element formulation of the implementation of the tetrahedral element. Numerous works on the development of FEM model of piezoelectric materials from 1970 to 2000 were reviewed by Benjeddou (2000). Over than 100 articles with different types of elements, i.e., solid, shell and beam elements, were developed during this period. Moreover, in the recent years, several advancements, i.e., Moving Kriging technique for mesh-free method (Bui et al. 2011), implementation on Abaqus User Element subroutine (Sartorato et al. 2015), Consecutive Interpolation technique with quadrilateral elements (Bui et al. 2016b), have also provided various alternatives on the piezoelectric structure analysis.

Despite numerous developments on the piezoelectric FEM model, almost all of them were implemented for actuator and sensor problems. There are only a few articles addressed energy harvesting purpose. The FEM formulation of piezoelectric energy harvester by De Marqui Jr. et al. (2009) is one of the most cited articles amongst those few. To the author’s knowledge, piezoelectric energy harvester model using IGA has not yet been developed. Furthermore, to this date, there are only a few published works presented the development of IGA models for piezoelectric structure.

Willberg and Gabbert (2012) proposed solid NURBS-based IGA piezoelectric elements. While Phung-Van et al. (2015), developed IGA shell for laminated composite plates with piezoelectric layers employing Higher-order Shear Deformation Theory (HSDT). Those models however limited to the application of sensors and actuators. Further development of piezoelectric IGA models, can be seen in the recent articles of Bui and colleagues. NURBS-based XIGA is implemented for crack problems on piezoelectric plates (Bui 2015) and piezoelectric embedded laminated composite (Bui et al. 2016a).

In the present work, high order IGA elements and FSDT are implemented to evaluate dynamic response from the laminated piezoelectric composite. The novel scheme coupled the structural responses of IGA and the voltage differential equations is proposed herein. Numerical experiments were carried out on the piezoelectric composite plate with various thickness. The discussion of the results and validation of the scheme are presented in some details.

## 2 Piezoelectric Energy harvester model

*D*, \(\varepsilon\),

*E*,

*T*,

*s*and

*S*are the electrical displacement (C/m\(^2\)), permittivity (F/m), electrical field (V/m), mechanical stress (N/m\(^2\)), compliance (m\(^2\)/N) and mechanical strain of the material. The piezoelectric charge constant,

*d*(m/V), denotes The coupling of the mechanical and electrical domain. It relates how much an electrical load, i.e., voltage, affect the mechanical deformation and vice versa.

- 1.
The electrical field only generated in z-direction \(E1=E2=0\ \text{and}\ E3\ne 0\);

- 2.
Only one mechanical stress components generated in the x-direction \(T2=\) ... \(T6=0\ \text{and}\ T1\ne 0\);

- 3.
All active layers are driven by the same voltage along z-direction,

*U*3.

Figure 1 shows a mechanical bending moment, *M* (Nm), is applied to the cantilever structure as well as the electrical resistance load, *R* (Ohm, Ω). Transverse displacement, *Z*, and bending slope, \({\partial Z }/{\partial x}\), are generated. In the active layer, the electrical field, *E*, and the electrical voltage, *U*, are created due to the structural displacement.

*Z*, is the combination of the displacement due to the mechanical load, \(Z_{mech}\), and the electrical load, \(Z_{elec}\).

*Q*(

*x*), from the root until a certain point at length

*x*, is expressed as a function of voltage input,

*U*(

*x*) and bending slope, \({\partial Z(x) }/{\partial x}\), with also material and geometrical properties of the beam/plate.

*b*and

*h*are the width and thickness of the piezoelectric layer, while \(h_{u}\) and \(h_{l}\) are the height of upper and lower surfaces of the layer to the neutral axis.

*I*is the electrical current (Ampere) and

*R*is the resistance load (Ohm, \({\varOmega }\)).

*t*and

*i*denote the time (s) and imaginary number, \(\sqrt{-\,1}\), with the geometrical and material parameters are defined by

## 3 NURBS and B-spline surface

*t*and

*s*with

*k*and

*l*orders.

*Q*(

*t*,

*s*) is the position of any point on the B-spline surface as a function of bi-parametric coordinate

*t*and

*s*. Meanwhile, \(B_{i,j}\) are the position vectors of the \(n+1\) control points at

*t*direction and \(m+1\) control points at

*s*direction. \(N_{i,k}(t)\) and \(M_{j,l}(s)\) are the normalized B-spline basis functions of order

*k*at

*t*direction and order

*l*at

*s*direction.

Physically, the B-spline surface function *Q*(*t*, *s*) represents the transformation of the real object/surface in the bi-parametric coordinate. The surface in Cartesian coordinate (*x*, *y*) is transformed into a surface that lies between *t* and *s* axes in the bi-parametric coordinate. Any points *Q* at the Cartesian coordinate is represented with any values of *t* and *s*.

A control net is formed by the control points. This control net functioned as the frame in which the B-spline surface follows its shape. Practically, these control points are the ones that changed or controlled in a technical drawing process (usually using a CAD program) to construct the desired shape of objects. While the level of smoothness or fairness controlled with the number of control points or the order level.

An example of B-spline surface constructed with a CAD program, Autocad\(^{\copyright }\), is shown in Fig. 2. Readers are refered to Shumaker et al. (2015), for details in NURBS or B-spline CAD modeling. Moreover, the algorithm samples of CAD’s spline model integration with finite element software, i.e. ABAQUS, can be seen in (Hammami et al. 2009; Lai et al. 2017b).

The shape and character of B-spline surface are also significantly influenced by the type of knot vectors used to construct the basis functions. The knot vectors are categorized into periodic and open type with uniformly spaced or non-uniformly spaced knot values (Rogers 2001). Periodic knot vectors contain knot values that increased from the start to the end. While open knot vectors has multiplicity at the start and at the ends equal to the order-*k* of the basis functions.

Example of a periodic uniform knot vector,

for \(k=3\), \({\mathbf {X}} = [0, \frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{5}{6}, 1]\).

Example of an open uniform knot vector,

for \(k=3\), \({\mathbf {X}} = [0, 0, 0, \frac{1}{3}, \frac{2}{3}, 1, 1, 1]\).

For the non-uniformly spaced knot vectors, the knot values may have unequal spacing and/or multiple internal knot values.

*i*,

*j*. In the case of all weighting values are equal to 1, the NURBS surface reduced to the standard B-spline surface. The addition of weighting values gives more flexibility to the creation of the surface. Thus, more complicated shape, i.e., circle or conic section, is conveniently constructible.

## 4 Isogeometric formulation for laminated composite

*R*is a point in the element with coordinate (

*x*,

*y*). While,

*nn*is the number of nodes in an element and \(N_{A}(t ,s)\) is the shape function of a particular node, \(R_{A}\), associated with the point

*R*.

## 5 Computational algorithm

- 1.
The displacement slope (bending angle) function due to the actual mechanical loading, \(\frac{\partial Z_{mech}}{\partial x}\). The load can be concentrated force, moment, or distributed pressure on the structure, or even a relative motion problem such as base excitation.

- 2.
The admittance function of bending angle due to a unit of moment, \(H_{\alpha m}\). The dummy load is given as a unit of moment on the neutral axis of the beam/plate tip. In the case of piezoelectric layers only cover part of the plate, the unit moment is given on the tip of the piezoelectric layer.

- 1.
The complex conjugate value of the bending angle due to the actual mechanical loading, \(\frac{\partial Z_{mech}}{\partial x}\).

- 2.
The admittance function, \(H_{\alpha m}\), obtained from the dummy load simulation.

- 3.
Material properties and geometrical information of the structure.

- 4.
Range of the resistance load,

*R*, used on the electrical circuit.

*b*(

*x*)).

In addition, it is worth to be highlighted, one of the main benefits of the hybrid scheme is the simplicity of its three primary processes. The two steps to simulate actual and dummy load simulation can be performed with standard structural analysis formulation/software, and the last step to calculate the voltage and power amplitude can be solved straightforwardly via Eqs. (16) and (17). Further elaborated in Sect. 6.2, the hybrid scheme could attain the same level of accuracy with faster computing time against fully coupled electromechanical FEM.

## 6 Validation

In this section, validation for the hybrid mathematical/numerical model is presented. The bimorph energy harvester model of Erturk and Inman (2011) under base excitation is used as analytical solution benchmark. The piezoelectric Kirchhoff plate of De Marqui Jr. et al. (2009) is used as benchmark for validation against electromechanically coupled finite element.

### 6.1 Validation against analytical solution

Material properties and geometry of the bimorph piezoelectric energy harvester (Erturk and Inman 2011)

Properties | Piezo ceramics | Host structure |
---|---|---|

Length | 30 | 30 |

Width | 5 | 5 |

Thickness | 0.15 (each) | 0.05 |

Material | PZT-5A | Aluminium |

Density \(\rho\) (kg/m\(^3\)) | 7750 | 2700 |

Elastic modulus \(1/S_{11}\) (GPa) | 61 | 70 |

Piezoelectric constant \(d_{31}\) (pm/V) | − 171 | – |

Permittivity \(\varepsilon _{33}\) (nF/m) | 15.045 | – |

The piezoelectric layers of bimorph energy harvester can be connected in series or parallel to the external resistance load. In the present work, series configuration of the bimorph is investigated. Three different methods, i.e., analytical, FEM and IGA are implemented to perform the structural dynamic analysis. Eight-noded bi-quadratic (Q8) FEM shell element, Nine-noded bi-quadratic (Q9) and 25-noded bi-quartic (Q25) IGA shell elements. Full integration points, \(3\times 3\) for bi-quadratic and \(5 \times 5\) for bi-quartic, are used. Computational codes are developed in MATLAB\(^{\copyright }\) to perform FEM and IGA structural dynamic analysis.

Natural frequency comparison, configuration from Table 1

Mode shape | Natural frequency (Hz) | |||
---|---|---|---|---|

Erturk–Inman | FEM Q8 | IGA Q9 | IGA Q25 | |

First bending | 185.1 | 187.3 | 187.2 | 187.3 |

Second bending | 1159.8 | 1172.5 | 1172.0 | 1172.5 |

Relative tip displacements and bending angles comparison, configuration from Table 1

Parameter | Relative tip displacement (μm) and bending angle (rad) | |||
---|---|---|---|---|

Erturk–Inman | FEM Q8 | IGA Q9 | IGA Q25 | |

Displacement | 78 | 78.3 | 78.3 | 78.3 |

Angle | 3.59e−3 | 3.61e−3 | 3.63e−3 | 3.62e−3 |

In Fig. 6a, b, denoted by “Present Model” are the present work results obtained via the hybrid mathematical/computational scheme. Denoted inside the brackets, “(FEM)”, “(IGA)” and “(Analytical)”, are the results of 3 different structural dynamic analysis coupled with the governing voltage function of Eq. (16). The FEM model used is of the FEM Q8 elements and the IGA model used is of the IGA Q25 elements.

The figures are in the logarithmic to logarithmic scale, where both the voltage and power amplitudes are normalized per unit of g (\(9.81\,\hbox {m/s}^2\)) and \(\hbox {g}^2\). The amplitude value “per g” means it is divided by the ratio of base acceleration to g, i.e. base amplitude \(1\,\upmu \hbox {m}\) and 185 Hz excitation equal with \(1.35\, \hbox {m/s}^2\) base acceleration or 0.14 acceleration ratio. Hence, 100 V at this acceleration is equal with 714.3 “V per g”. In addition, the electrical power is normalized per unit \(\hbox {g}^2\), or the power amplitude divided by the square of the acceleration ratio.

Electrical parameters comparison, configuration from Table 1

Parameter | Electrical parameters comparison | |||
---|---|---|---|---|

Erturk–Inman | Present (analyt.) | Present (FEM) | Present (IGA) | |

Max voltage (V) | 0.78 | 0.78 | 0.79 | 0.79 |

Max power (μW) | 4.35 | 4.35 | 4.43 | 4.43 |

R at max power (kΩ) | 35.9 | 35.9 | 36.11 | 36.09 |

### 6.2 Validation against electromechanically coupled FEM

Material properties and geometry of the bimorph UAV Wingspar (De Marqui Jr. et al. 2009)

Dimension of the beam | |

Total length | 300 |

Total width | 30 |

Total thickness | 12 |

Piezoceramics (PZT-5A) | |

| 7800 |

\(c_{11}^{E}\), \(c_{22}^{E}\) (GPa) | 120.3 |

\(c_{12}^{E}\) (GPa) | 75.2 |

\(c_{13}^{E}\), \(c_{23}^{E}\) (GPa) | 75.1 |

\(c_{33}^{E}\) (GPa) | 110.9 |

\(c_{66}^{E}\) (GPa) | 22.7 |

\(e_{31}\), \(e_{32}\) (C/m | − 5.2 |

\(e_{33}\) (C/m | 15.9 |

\(\varepsilon _{33}\) (nF/m) | 15.93 |

Host-structure (aluminum) | |

| 2750 |

| 70 |

In Table 5, the piezoelectric properties are shown in 3D orthotropic of the stress-charge form (Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society 1988). The \(c_{11}^{E},c_{12}^{E},\ldots ,c_{66}^{E}\) are the components of elastic stiffness of the mechanical constitutive relations. The \(e_{31},e_{32},e_{33}\) are the components of piezoelectric charge constant.

De Marqui Jr. et al. (2009) adopted Kirchhoff plate theory to develop the electromechanically coupled finite element model of the piezoelectric energy harvester. The piezoceramic layer is assumed poled in the thickness direction thus align with the assumption used at 2. The piezoceramic layer is also assumed covered by a continuous electrode and perfectly bonded to the host-structure. It is assumed very thin and conductive electrode layers are on the top and bottom surfaces of the piezoceramic layers. Hence, it is assumed that all finite elements generate the same voltage output. Furthermore, one degree of freedom is used as the voltage output degree of freedom of each element. The interested reader is referred to De Marqui Jr. et al. (2009) for the detail of the formulation.

It is seen from Fig. 8 the trends of the natural frequency and damping ratio for all three procedures are all in good comparison. The total mass of the beam is fixed at 1.1 full aluminum mass, and the cross section shape is maintained. It is, therefore, the only variation that affecting the change of the natural frequency is the composite material properties along the span.

In general, the natural frequency is decreasing from \(\hbox {L}^{*} = 0.1\) to 1. However, from \(\hbox {L}^{*}\) = 0.25 to 0.4 a slight increment occurs, despite it is followed by a significant drop from \(\hbox {L}^{*}\) = 0.8 to 1. This trend is caused by a reduced stiffness of the piezo-aluminum composite beam as the piezoceramics layers approaches \(\hbox {L}^{*} = 1\). In addition, for a benchmark, analytically the natural frequency of a full aluminum spar is 108.79 Hz.

The maximum power and the resistance load at the maximum power for various length ratio are depicted in Fig. 9a, b. All of the three procedures are also in good agreement. The maximum power increased at first and reached its highest point at \(\hbox {L}^{*} = 0.225\) before it significantly drops when \(\hbox {L}^{*}\) approach 1. In contrast, the resistance load at the maximum power is decreased with the increment of \(\hbox {L}^{*}\).

Natural frequency and maximum power comparison of De Marqui Jr. et al.---FEM and present model (FEM)

\(\hbox {L}^*\) | \(\hbox {h}^*\) | \(\hbox {F}_{DMJ}\,\hbox {(Hz)}\) | \(\hbox {F}_{present}\,(Hz)\) | ΔF (%) | \(\hbox {P}_{DMJ}\, (\hbox {mW/g}^{2}\)) | \(\hbox {P}_{present} (\hbox {mW/g}^{2}\)) | ΔP (%) |
---|---|---|---|---|---|---|---|

0.1 | 0.272 | 107.36 | 106.78 | 0.54 | 154.00 | 152.79 | 0.79 |

0.2 | 0.136 | 106.97 | 106.39 | 0.54 | 157.69 | 159.06 | 0.87 |

0.225 | 0.121 | 106.97 | 106.40 | 0.53 | 157.72 | 159.24 | 0.97 |

0.25 | 0.109 | 106.98 | 106.42 | 0.52 | 157.65 | 159.18 | 0.97 |

0.4 | 0.068 | 107.10 | 106.59 | 0.48 | 155.38 | 156.88 | 0.96 |

0.5 | 0.054 | 107.04 | 106.56 | 0.45 | 151.90 | 153.22 | 0.86 |

0.6 | 0.045 | 106.79 | 106.33 | 0.43 | 146.27 | 147.33 | 0.74 |

0.8 | 0.034 | 105.55 | 105.12 | 0.41 | 127.56 | 127.59 | 0.02 |

1.0 | 0.027 | 103.34 | 102.95 | 0.38 | 103.22 | 102.79 | 0.41 |

Natural frequency and maximum power comparison of De Marqui Jr. et al.---FEM and present model (IGA)

\(\hbox {L}^*\) | \(\hbox {h}^*\) | \(\hbox {F}_{DMJ}\,(Hz)\) | \(\hbox {F}_{present}\,(Hz)\) | ΔF (%) | \(\hbox {P}_{DMJ}\,(\hbox {mW/g}^{2}\)) | \(\hbox {P}_{present}\,(\hbox {mW/g}^{2}\)) | ΔP (%) |
---|---|---|---|---|---|---|---|

0.1 | 0.272 | 107.36 | 107.05 | 0.28 | 154.00 | 152.07 | 1.25 |

0.2 | 0.136 | 106.97 | 106.67 | 0.28 | 157.69 | 158.92 | 0.78 |

0.225 | 0.121 | 106.97 | 106.61 | 0.33 | 157.72 | 158.50 | 0.50 |

0.25 | 0.109 | 106.98 | 106.70 | 0.26 | 157.65 | 158.45 | 0.50 |

0.4 | 0.068 | 107.10 | 106.85 | 0.24 | 155.38 | 156.50 | 0.47 |

0.5 | 0.054 | 107.04 | 106.77 | 0.25 | 151.90 | 152.63 | 0.72 |

0.6 | 0.045 | 106.79 | 106.56 | 0.22 | 146.27 | 147.26 | 0.66 |

0.8 | 0.034 | 105.55 | 105.37 | 0.17 | 127.56 | 127.32 | 0.19 |

1.0 | 0.027 | 103.34 | 103.23 | 0.11 | 103.22 | 102.33 | 0.86 |

The comparison of The present hybrid scheme with FEM Q8 combination is provided in Table 6. It is shown that the natural frequencies and maximum power amplitude are all in good agreement with variances less than 1%. Moreover, the results of IGA Q25 couple is shown in Table 7 with also insignificant variations compared to De Marqui Jr. et al results. These results demonstrate the robustness of the present hybrid scheme, the capability to estimate energy harvested from a structure with non-uniform material properties and obtained a good level of accuracy similar to the electromechanically coupled finite element model.

Simulation time comparison

Simulation time (s) | |||||
---|---|---|---|---|---|

Steps | 100 no. of R | 10,000 no. of R | |||

Hybrid present | FEM DMJ | Hybrid present | FEM DMJ | ||

A. | Non-coupled simulations | 2 × 15 | – | 2 × 15 | – |

B. | \(\bar{U}\) and \(P_{max}\) calculations | 5 | – | 15 | – |

C. | Fully coupled simulations | – | 100 × 20 | – | 10,000 × 20 |

Total | 35 | 2000 | 45 | 20,000 |

In Table 8, “Hybrid-Present” denotes the present hybrid scheme coupled with IGA, while the full electromechanically coupled FEM of De Marqui Jr. et al. (2009) is denoted with “FEM–DMJ”. As explained in Sect. 5 and Fig. 3, the present hybrid scheme only requires three primary processes consisted of two numerical simulations for actual and dummy load, and one process to calculate voltage-power harvested. Step A denotes the non-coupled/purely mechanical loads numerical simulations performed via IGA Q25 configuration. Calculation of voltage (\(\bar{U}\)) and power (\(P_{max}\)) for N-number of resistance loads,*R*, is denoted by step B. Process of calculation in step B is also performed via a MATLAB\(^{\copyright }\) computational code.

For “FEM–DMJ” simulation, the only step used is step C, the full electromechanically coupled finite element simulation. In step C, the resistance load and the mechanical load are both given as the excitation source on the finite elements, while both the structural deformation and voltage responses are obtained directly as the output of FEM simulation. It is assumed that for a set of N-numbers of R, N-times of simulations is required.

As shown in Table 8, step A shows fixed simulation time, \(2\times 15\) s. The elapsed time for the actual load (base excitation) and the dummy load (the moment at the tip) are around 15 s each and independent to the number of R. In step B, shows an increment of computing time as the number of resistance loads is increasing. For a set of 100 variances of R, the elapsed time is less than 5 s. While for 10,000 numbers of R, around 15 s computing time is required. Therefore, in total, “Hybrid–Present” simulations require 35 s for 100 variances of R, and 45 s for 10,000 numbers of R.

In contrast, the simulation time required for “FEM–DMJ”, step C, is purely dependent on the number of R observed. For a simulation with one resistance load, it only requires less than 20 s. However, the simulation times are multiplied with N-number of R investigated. Therefore, if 100 number of R used, then 2000 s is required. Furthermore, if 10,000 number of R used, then 200,000 s or around 56 h is required. Interesting to note that to produce a plot with the level of detail such as shown in Fig. 10, a set of R from \(10^{2}\) to \(10^{8}\,{\varOmega }\) with 100 \({\varOmega }\) step is used. Thus, to produce this plot utilizing the present hybrid scheme only need less than 1 min, while with full electromechanically coupled FEM will require around 56 h.

The independence on the number of simulations on the hybrid scheme is beneficial especially in a preliminary/conceptual design stage, in which a faster iterative design process to obtain an optimal harvester structural design and resistance load is achievable. However, despite the higher computational cost, a full electromechanically coupled FEM may provide more details for a particular area of interests, i.e., the region near optimum resistance load. Thus, the hybrid scheme may build the fundamental sense of the best harvester design at the early design stage, while more detailed analysis may be provided by fully coupled FEM at the later design phase. Further in Sect. 7, the robustness of the present hybrid scheme with IGA combination is elaborated with very thin structural configurations.

## 7 Case study of plates with various thickness

As mentioned in Sect. 1, Thai et al. (2012) implemented NURBS-based IGA to investigate the shear locking phenomena on the dynamic response problems of the laminated composite plate. In this section, a new investigation on the effect of shear locking phenomena and the resistance of high-order IGA elements to the voltage and power responses of piezoelectric energy harvester is presented.

Similar procedure as presented in Sect. 6.1 is implemented. FEM Q8, IGA Q9, and IGA Q25 elements are deployed. The same number of elements, \(2 \times 12\) equal sized elements, is used. The thickness ratio of the beam is varied from \(\hbox {h}_0/\hbox {h} = 1\) to \(\hbox {h}_0/\hbox {h} = 10^{4}\). As the thickness ratio, \(\hbox {h}_0/\hbox {h}\) increases, the investigated thickness becomes smaller. At \(\hbox {h}_0/\hbox {h} = 10^{4}\), the length per thickness ratio of the element is more than 7000 and considered as a very thin element.

For the finite element results, at the very thin plates, the voltage response is underestimated at the range of resistance load close to short circuit (\(R \rightarrow 0\)). The geometrical configuration and material properties, \({\varGamma }_{1}\, \& \,{\varGamma }_{2}\), unaffected by the numerical results. However, for the finite element results at very thin plates, the reverse piezoelectric effect is decreased, thus, at the range near short circuit, the reverse piezoelectric parameter unable to overcome the 1/*R* parameter. Moreover, with also smaller mechanical deformation, hence the voltage response is underestimated.

*R*parameter is decreased. Furthermore, As for FEM result, the reverse piezoelectric also dropped at the very thin plate, it led to the overestimated voltage response.

Electrical parameters comparison with h\(_0\)/h = 10\(^{4}\)

Parameter | Electrical parameters comparison | |||
---|---|---|---|---|

Erturk Inman | Present (analyt.) | Present (FEM) | Present (IGA) | |

Max voltage (nV) | 7.8 | 7.8 \({\varDelta }=0.0\%\) | 14.5 \({\varDelta }=85.9\%\) | 7.64 \({\varDelta }=2.0\%\) |

Max power (pW) | 4.35e−10 | 4.35e−10 \({\varDelta }=0.0\%\) | 4.55e−10 \({\varDelta }=4.6\%\) | 4.17e−10 \({\varDelta }=4.1\%\) |

R at max power (kΩ) | 35.9 | 35.9 \({\varDelta }=0.0\%\) | 125.07 \({\varDelta }=248\%\) | 36.09 \({\varDelta }=0.5\%\) |

In detail, Table 9 presented the comparison of maximum voltage and maximum power amplitude with the resistance at maximum power for all procedures at \(\hbox {h}_0/\hbox {h} = 10^{4}\). It can be seen that all of the methods are in good agreement except the one combined with FEM. Although the maximum power obtained with FEM combination is less than 5% variance (\({\varDelta }\)), however the resistance value at the maximum power is overestimated by 2.5 times the analytical result.

## 8 Conclusions

The hybrid mathematical model/Isogeometric Analysis (IGA) scheme to evaluate a cantilevered piezoelectric energy harvester under dynamic bending has been developed. The hybrid scheme has shown beneficial advantages, i.e., the simplicity of the primary processes on the use of standard structural analysis and straightforward governing voltage equation. It is shown in this paper; the hybrid scheme reduced computational cost compared to the more sophisticated electromechanically coupled finite elements. The capabilities and robustness of the procedure are demonstrated by comparison with results from the literature.

The numerical investigations focused on the shear locking resistance of the higher-order IGA elements have been performed. It is shown in this paper; the higher-order IGA elements are less prone to the shear locking phenomena. Employing higher-order IGA, the structural deformations and the energy responses are maintained at a reasonable level of accuracy even at a very thin structure. It is considered advantageous for the analysis of piezoelectric materials, in which usually manufactured as thin-walled structures.

However, as it is discussed earlier in the Introduction section, one of the gaps in the piezoelectric energy harvesting study is the lack of models for more general loading conditions and complex structure. Despite the ability of the hybrid formulation to provide analysis on various form of loads, i.e., force, pressure, base excitation, it is still limited to the dynamic bending cases. In order to obtain more general formulation, mixed mechanical loads, i.e., bending and torsion, need to be considered in the further development. Further elaboration on more complicated structural configuration, i.e., a tapered section, twisted section, may as well required. In this case, the ability of NURBS-based IGA on the integration with CAD software and complex shapes construction are worth to be explored in the future works.

Concerning higher-order NURBS-based IGA, it is found from the literature that it may have a negative effect on the computational cost. For a fixed number of unknown, higher-order NURBS-based IGA may consume more CPU time and RAM (Collier et al. 2012). Therefore, it is considered worth to be investigated in the future work the effect of the higher-order NURBS-based IGA to the overall computational cost of the present hybrid scheme. Despite in this paper, it is shown that there is no significant issue with the computing time, investigation on the more complex structure and more dense mesh may need to be performed in the future.

It is also found that the use of the mass matrices in higher-order IGA for structural vibration analysis is still not optimal (Cottrell et al. 2006). The lumped mass matrices limit the accuracy to just second order, even for higher-order of IGA elements. However, some attempts to develop novel high-order mass matrices are found in the literature (Wang et al. 2013, 2015). Those new mass matrices are proven to increase the level of accuracy. Thus, it may be used to enhance the performance of the present IGA elements.

## Notes

### Acknowledgements

The authors gratefully acknowledge Dr. Anton Krynkin for the fruitful discussions and the support of Indonesia Endowment Fund for Education (LPDP).

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